Lecture Notes 10.5: EM Waves in Resonant Cavities

UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 10.5 Prof. Steven Errede

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved.

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LECTURE NOTES 10.5

EM Standing Waves in Resonant Cavities

One can create a resonant cavity for EM waves by taking a waveguide (of arbitrary shape) and closing/capping off the two open ends of the waveguide.

Standing EM waves exist in (excited) resonant cavity (= linear superposition of two counter-propagating traveling EM waves of same frequency).

Analogous to standing acoustical/sound waves in an acoustical enclosure. Rectangular resonant cavity – use Cartesian coordinates Cylindrical resonant cavity – use cylindrical coordinates to solve the EM wave eqn. Spherical resonant cavity – use spherical coordinates

A.) Rectangular Resonant Cavity: ( L W H a b d ) with perfectly conducting walls (i.e. no dissipation/energy loss mechanisms present), with 0 x a , 0 y b , 0 z d . n.b. Again, by convention: a > b > d.

Since we have rectangular symmetry, we use Cartesian coordinates - seek monochromatic EM plane wave type solutions of the general form:

, , , , ,

, , , , ,

i to

i to

E x y z t E x y z e

B x y z t B x y z e

Maxwell’s Equations (inside the rectangular resonant cavity – away from the walls):

(1) Gauss’ Law: (2) No Monopoles:

0E 0oE

0B 0oB

(3) Faraday’s Law: (4) Ampere’s Law:

B

Et

o oE i B

2

1 EB

c t

2o oB i E

c

Take the curl of (3): = 0 {Gauss’ Law}

2

2o o o o oE i B E E i B E

c

{using (4) Ampere’s Law}

22

22

22

ox ox

oy oy

oz oz

E Ec

E Ec

E Ec

, ,

, , . . each is a fcn , ,

, ,

ox ox

oy oy

oz oz

E E x y z

E E x y z i e x y z

E E x y z

Subject to the boundary conditions

|| 0E and 0B

at all inner surfaces.



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For each component , , of , ,ox y z E x y z we try product solutions and then use the

separation of variables technique:

, ,io i i iE x y z X x Y y Z z for

22 , ,

i io oE x y z Ec

where subscript , ,i x y z .

22 2 2

2 2 2i i i

i i i i i i i i i

X x Y y Z z cY y Z z X x Z z X x Y y X x Y y Z z

x y z c

Divide both sides by i i iX x Y y Z z :

The wave equation becomes:

22 2 2

2 2 2

only only only

1 1 1i i i

i i i

fcn x fcn y fcn z

X x Y y Z z

X x x Y y y Z z z c

This equation must hold/be true for arbitrary (x, y, z) pts. interior to resonant cavity

0

0

0

x a

y b

z d

This can only be true if:

2

22

1constanti

xi

X xk

X x x

2

22

0ix i

X xk X x

x

22

2

1 constanti

yi

Y yk

Y y y

2

22

0iy i

Y yk Y y

y

22

2

1 constanti

zi

Z zk

Z z z

2

22

0iz i

Z zk Z z

z

with: 2

2 2 2 2x y zk k k k

c

characteristic equation

General solution(s) are of the form: , , :i x y z

, , cos sin cos sin cos sinio i x i x i y i y i z i zE x y z A k x B k x C k y D k y E k z F k z

n.b. In general, , and x y zk k k should each have subscript , , ,i x y z but we will shortly find out

that ixk same for all , , ,

iyi x y z k same for all , , ,i x y z and izk same for all , ,i x y z .

n.b. We seek oscillatory

(not damped) solutions !!!



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Boundary Conditions: || 0E @ boundaries and 0B @ boundaries:

0, 0 at

0, ox

y y bE

z z d

coefficients 0

0

x

x

E

C

and , 1, 2,3,....

, 1, 2,3,....

y

z

k n b n

k d

0, 0 at

0, oy

x x aE

z z d

coefficients 0

0

y

y

A

E

and , 1, 2,3,....

, 1, 2,3,....x

z

k m b m

k d

0, 0 at

0, oz

x x aE

y y b

coefficients 0

0

z

z

A

C

and , 1, 2,3,....

, 1, 2,3,....x

y

k m a m

k n d n

n.b. m = 0, and/or n = 0 and/or 0 are not allowed, otherwise , , 0ioE x y z (trivial solution).

Thus we have (absorbing constants/coefficients, & dropping x,y,z subscripts on coefficients):

, , cos sin sin sin

, , sin cos sin sin

, , sin sin cos sin

ox x x y z

oy x y y z

ox x y z z

E x y z A k x B k x k y k z

E x y z k x C k y D k y k z

E x y z k x k y E k z F k z

But (1) Gauss’ Law: 0E 0oyox oz

EE E

x y z

Thus:

sin cos sin sin

sin sin cos sin

sin sin sin cos 0

x x x y z

y x y y z

z x y z z

k A k x B k x k y k z

k k x C k y D k y k z

k k x k y E k z F k z

This equation must be satisfied for any/all points inside rectangular cavity resonator. In particular, it has to be satisfied at , ,x y z 0,0,0 .

We see that for the locus of points associated with (x = 0,y,z) and (x,y = 0,z) and (x,y,z = 0),

we must have 0B D F in the above equation.

Note also that for the locus of points associated with 2 , ,xx m k y z and , 2 ,yx y n k z

and , , 2 zx y z k where , ,m n odd integers (1, 3, 5, 7, etc. …) we must have:

0x y zAk Ck Ek .

Note further that this relation is automatically satisfied for , ,m n even integers (2, 4, 6, 8, etc. …).



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Thus:

1, 2,3, 4,, , cos sin sin

, , sin cos sin 1, 2,3, 4,

, , sin sin cos 1, 2,3, 4,

x

ox x y z

oy x y z y

oz x y z

z

mk m

aE x y z A k x k y k zn

E x y z C k x k y k z k nb

E x y z E k x k y k zk

d

With: ˆ ˆ ˆ, ,o ox oy ozE x y z E x E y E z n.b.: 0m n simultaneously is not allowed!

Now use Faraday’s Law to determine B

:

iB E

sin cos cos sin cos cosoyozox y x y z z x y z

EEi iB Ek k x k y k z Ck k x k y k z

y z

cos sin cos cos sin cosox ozoy z x y z x x y z

E Ei iB Ak k x k y k z Ek k x k y k z

z x

cos cos sin cos cos sinoy oxoz x x y z y x y z

E Ei iB Ck k x k y k z Ak k x k y k z

x y

or:

ˆ ˆ ˆ, ,

ˆ sin cos cos

ˆ cos sin cos

ˆ cos cos sin

o ox oy oz

y z x y z

z x x y z

x y x y z

B x y z B x B y B z

iEk Ck k x k y k z x

Ak Ek k x k y k z y

Ck Ak k x k y k z y

This expression for , ,oB x y z (already) automatically satisfies boundary condition (2) 0B :

0oxB at 0,x x a 0oyB at 0,y y b 0ozB at 0,z z d

with x

mk

a

with y

nk

b

with zkd

0,1, 2,m 0,1, 2,n 0,1, 2,



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Does , , 0oB x y z ???

, ,

cos cos cos

oyox ozo

x y z x y z x y

BB BB x y z

x y z

i ik Ek Ck k x k y k z k k E

x zk k C

cos cos cos y z x x y z y zk Ak Ek k x k y k z k k A x yk k E

cos cos cos z x y x y z x zk Ck Ak k x k y k z k k C y zk k A

cos cos cosx y zk x k y k z

, , 0oB x y z YES!!!

For TE Modes:

0zE

coefficient 0E . Then 0x y zAk Ck Ek tells us that: 0x yAk Ck or: x

y

kC A

k

Thus:

, , , cos sin sin i tox x y zE x y z t A k x k y k z e , 1, 2,3,x

mk m

a

, , , sin cos sin i txoy x y z

y

kE x y z t A k x k y k z e

k

, 1, 2,3,y

nk n

b

, , , 0ozE x y z t , 1, 2,3,zkd

(n = 0 is NOT allowed for TE modes!!!)

, , , sin cos cos i txox z x y z

y

kiB x y z t A k k x k y k z e

k

, , , cos sin cos i toy z x y z

iB x y z t k A k x k y k z e

, , , cos cos sin i txoz x y x y z

y

kiB x y z t A k k k x k y k z e

k

The angular cutoff frequency for th, ,m n mode for TE modes in a rectangular cavity is:

2 2 2

mn

m nc

a b d

and: gv c vk

no dispersion.

The lowest

, ,m nTE

mode

a b d

is: 111TE



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For TM Modes:

0zB

0x yCk Ak or: y

x

kC A

k

Thus:

, , , cos sin sin i tox x y zE x y z t A k x k y k z e , 1, 2,3,x

mk m

a

, , , sin cos cosy i toy x y z

x

kE x y z t A k x k y k z e

k

, 1, 2,3,y

nk n

b

(m = 0 is NOT allowed for TM modes!!!) , 1, 2,3,zkd

, , , sin sin cosy y i txoz x y z

z x z

k kkE x y z t A k x k y k z e

k k k

, , , sin cos cosy y y i tox x y z x y z

x z x

k k kiB x y z t A k k A k k x k y k z e

k k k

, , , cos sin cosy i txoy z x y x y z

x x

k kiB x y z t Ak A k k k x k y k z e

k k

, , , 0ozB x y z t

For either TE or TM modes: 2

2 2 2 2x y zk k k k

c

with:

, 1, 2,x

mk m

a

, 1, 2,y

nk n

b

, 1,2,zkd

The angular cutoff frequency for th, ,m n mode is the same for TE/TM modes in a rectangular cavity:

2 2 2

mn

m nc

a b d

and: gv c vk

no dispersion.

The lowest

, ,m nTM

mode

a b d

is: 111TM



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B.) The Spherical Resonant Cavity:

The general problem of EM modes in a spherical cavity is mathematically considerably more

involved (e.g. than for the rectangular cavity) due to the vectorial nature of the and E B

-fields. For simplicity’s sake, it is conceptually easier to consider the scalar wave equation, with a

scalar field ,r t satisfying the free-source wave equation 2

22

,1, 0

r tr t

c t

which can be Fourier-analyzed in the complex time-domain , , i tr t r e d

with each Fourier component ,r satisfying the Helmholtz wave equation:

2 2 , 0k r

with: 22k c i.e. no dispersion.

In spherical coordinates, the Laplacian operator is:

222

2 2 2 2 2

, ,1 1 1, , sin

sin sin

r rr r r

r r r r

To solve this scalar wave equation – we again try a product solution of the form:

, i tR rr P Q e

r

,sphericalharmonics

,m mm

f r Y

Plug this ,r into the above scalar wave equation, use the separation of variables technique:

Get radial equation:

22

2 2

120

d dk f r

dr r dr r

where: = 0, 1, 2, 3, . . .

Let: 1f r u r

r . Then we obtain Bessel’s equation with index 1

2v :

22 1

222 2

10

d dk u r

dr r dr r

Solutions of the (radial) Bessel’s equation are of the form: 1 12 2

1 12 2

Bessel fcn of 1st Bessel fcn of 2ndkind of order kind of order

m mm

A Bf r J kr N kr

r r

It is customary to define so-called spherical Bessel functions and spherical Hankel functions:

12

122

j x J xx

where: x kr

12

122

n x N xx

The ,mY satisfy

the angular portion of scalar wave equation…



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and: 12

1 12 2

1,2 2

h x J x iN x j x i n xx

sin 1

xdj x x

x dx x

cos1 xdn x x

x dx x

0

sin

xj x

x 1

0 ixe

h xix

0

cos xn x

x 2

0

ixeh x

ix

1 2

sin cos

x xj x

x x 1

1 1ixe i

h xx x

1 2

cos sinx xn x

x x 2

1 1ixe i

h xx x

For 1, x :

2

1 ...2 1 !! 2 2 3

x xj x

where: 2 1 !! 2 1 2 1 2 3 ... 5 3 1

2

1

2 1 !!1 ...

2 1 2

xn x

x

For 1, x :

1 sin

2j x x

x

1cos

2n x x

x

The general solution to Helmholtz’s equation in spherical coordinates can be written as:

1 1 2 2

,

, ,m m mm

r t A h kr A h kr Y

Coefficients are determined by boundary conditions.

For the case of EM waves in a spherical resonant cavity we will (here) only consider TM modes,

which for spherical geometry means that the radial component of , 0rB B

. We further assume

(for simplicity’s sake) that the E

and B

-fields do not have any explicit -dependence.

n.b. If x = kr is real, then 2 * 1h x h x



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Hence: 2 1 !

, cos4 !

m imm

mY P e

m

Will have some restrictions imposed on it Associated Legendré Polynomial

If 0rB and B

explicit function of , then: 0B 0B {necessarily}

But: B

Et

requires: 0E

TM modes with no explicit -dependence involve only rE , E and B

Combining B

Et

and 2

1 EB

c t

with harmonic time dependence i te of solutions,

We obtain: 2

0B Bc

The -component of this equation is:

2 2

2 2

1 1sin 0

sinrB rB rB

c r r

But: 2

~Legendré equation with 1

1 1sin sin

sin sin sin

m

rB rBrB

Try product solutions of the form: 1, cosu r

B r Pr

Substituting this into the above equation gives a differential equation for u r of the form of:

Bessel’s equation:

22

2 2

10

d u ru r

dr c r

with = 0, 1, 2, 3, . . . defining the

angular dependence of the TM modes.

Let us consider a resonant spherical cavity as two concentric, perfectly conducting spheres of inner radius a and outer radius b.

If 1, cosu r

B r Pr

, the radial and tangential electric fields (using Ampere’s Law) are:

2 2

, sin 1 cossinr

u ric icE r B P

r r r

2 2

1, cosu ric ic

E r rB Pr r r r



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But E E which must vanish at r = a and r = b

0r a r b

u r u r

r r

The solutions of the radial Bessel equation are spherical Bessel functions (or spherical Hankel functions).

The above radial boundary conditions on

0r ar b

u r

r

lead to transcendental equations for

the characteristic frequencies, {eeeEEK}!!!

However {don’t panic!}, if: (b – a) = h is such that h a then:

2 2

1 1constant!!!

r a

And thus in this situation, the solutions of Bessel’s equation:

22

2 2

10

d u ru r

dr c a

2

22

0d u r

k u rdr

where:

22

2

1k

c a

are simply sin (kr) and cos (kr) !!! i.e. cos sinu r A kr B kr

Then: sin cos 0r a

ukA ka kB ka

r

and sin cos 0

r b

ukA kb kB kb

r

For b a h a an approximate solution is: cosu r A kr ka

with: kh k b a n , n = 0, 1, 2, . . .

Thus: 2 2

22

1n

nk

c a h

, n = 0, 1, 2, 3, . . . and = 0, 1, 2, 3, . . .

The corresponding angular cutoff frequency is:

22

2 2

1 1n n

nc k c

a h a

for h a , n = 0, 1, 2, 3, . . . and = 0, 1, 2, 3, . . .

Because h a , we see that the modes with n = 1, 2, 3, . . . turn out to have relatively high

frequencies n

nc

h

for 1n . However, the n = 0 modes have relatively low frequencies:

0 2

11

cc

a a

for h a .

An exact solution (correct to first order in (h/a) expansion) for n = 0 is: 0 1

2

1c

a h

These eigen-mode frequencies are known as Schumann resonance frequencies. = 1, 2, 3, . . . (W.O. Schumann – Z. Naturforsch. 72, 149, 250 (1952))



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For n = 0, the EM fields are: 0 0E , 02

1cosrE P

r

and 0 11cosB P

r

Very Useful Table:

ˆ ˆr ˆ ˆr ˆ ˆ r ˆˆ r

ˆˆ r ˆˆr Poynting’s vector:

1 10 0 0 3 3

0

1 1 1 ˆˆˆ cos cos cos cosS E B r P P P Pr r

The Earth’s surface and the Earth’s ionosphere behave as a spherical resonant cavity (!!!) with the Earth’s surface {approximately} as the inner spherical surface: 6378 a r r km

66.378 10 m (= Earth’s mean equatorial radius), the height h (above the surface of the Earth) of the ionosphere is: 5100 10h km m ( a ) → b = a + h 6.478 x 106 m.

For the n = 0 Schumann resonances: 0 1

2

1c

a h

for h a .

1: 01 1

2

2c

a h

01

01 10.5 2

f Hz

2 : 02 1

2

6c

a h

02

02 18.3 2

f Hz

3 : 03 1

2

12c

a h

03

03 25.7 2

f Hz

4 : 04 1

2

20c

a h

04

04 33.2 2

f Hz

5 : 05 1

2

30c

a h

05

05 46.7 2

f Hz

(. . . etc.)

The n = 0 Schumann resonances in the Earth-ionosphere cavity manifest themselves as peaks in the noise power spectrum in the VLF (Very Low Frequency) portion of the EM spectrum → VLF EM standing waves in the spherical cavity of the Earth-ionosphere system!!!

y

z

x

0ˆ ˆ, B 0ˆ ˆ, rr E r

0S

a

b

Circumpolar N-S waves!

n.b. For the n = 1 Schumann resonances:

1 1.5 f KHz



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Schumann resonances in the Earth-ionosphere cavity are excited by the radial E

-field component of lightning discharges (the frequency component of EM waves produced by lightning at these Schumann resonance frequencies).

Lightning discharges (anywhere on Earth) contain a wide spectrum of frequencies of EM radiation – the frequency components f01, f02, f03, f04, . . excite these resonant modes – the Earth literally “rings like a bell” at these frequencies!!! The n = 0 Schumann resonances are the lowest-lying normal modes of the Earth-ionosphere cavity.

Schumann resonances were first definitively observed in 1960. (M. Balser and C.A. Wagner, Nature 188, 638 (1960)).

Nikola Tesla may have observed them before 1900!!! (Before the ionosphere was known to even exist!!!) He also estimated the lowest modal frequency to be f01 ~ 6 Hz!!! n = 0 Schumann Resonances:



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The observed Schumann resonance frequencies are systematically lower than predicted,

(primarily) due to damping effects: 2 20

11

i

Q

where Q = Quality factor 0

= “Q”

of resonance, and = width at half maximum of power spectrum:

The Earth’s surface is also not perfectly conducting. Seawater conductivity 0.1C Siemens!!

Neither is the ionosphere! → Ionosphere’s conductivity 4 710 10C Siemens

On July 9, 1962, a nuclear explosion (EMP) detonated at high altitude (400 km) over Johnson Island in the Pacific {Test Shot: Starfish Prime, Operation Dominic I}. - Measurably affected the Earth’s ionosphere and radiation belts on a world-wide scale! - Sudden decrease of ~ 3 – 5% in Schumann frequencies – increase in height of ionosphere!

- Change in height of ionosphere: 2 0.03 0.05 400 600 h h h R km !!!

- Height changes decayed away after ~ several hours. - Artificial radiation belts lasted several years!

Note that # of lightning strikes, (e.g. in tropics) is strongly correlated to average temperature. Scientists have used Schumann resonances & monthly mean magnetic field strengths to monitor lightning rates and thus monitor monthly temperatures – they all correlate very well!!!

Monitoring Schumann Resonances → Global Thermometer → useful for Global Warming studies!!

Earth Coordinate System:

0 12

12

cf

a h

0 0E (north – south)

02

1cosrE P

r

(up – down)

0 11cosB P

r (east – west)

10 3

1 ˆcos cosS P Pr

(north – south)

For the n = 0 modes of Schumann Resonances:

ˆE r (up – down) ˆB

(east – west) ˆS

(north – south)



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We can observe Schumann resonances right here in town / @ UIUC!! Use e.g. Gibson P-90 single-coil electric guitar pickup 90 10 Henrys, ~10K turns #42AWG copper wirePL for

detector of Schumann waves and a spectrum analyzer (e.g. HP 3562A Dynamic Signal Analyzer) – read out the HP 3562A into PC via GPIB.

Orientation/alignment of Gibson P-90 electric guitar pickup is important – want axis of

pickup aligned ˆB

(i.e. oriented east – west) as shown in figure below. n.b. only this orientation yielded Schumann-type resonance signals {also tried 2 other 90o orientations {up-down} and {north-south} but observed no signal(s) for Schumann resonances for these.}

Electric guitar PU’s are very sensitive – e.g. they can easily detect car / bus traffic on street below from 6105 ESB (6th Floor Lab) – can easily see car/bus signal from PU on a ‘scope!!! n.b. PU housed in 4 closed, grounded aluminum sheet-metal box to suppress electric noise.



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Lecture Notes 10.5: EM Waves in Resonant Cavities

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